1. N-way ANOVA

2. Two-factor ANOVA with equal replications Experimental design: 2  2 (or 2 2 ) factorial with n = 5 replicate Total number of observations: N = 2  2  5 = 20 Equal replications also termed orthogonality 2

3. The hypothesis H 0 : There is on effect of hormone treatment on the mean plasma concentration H 0 : There is on difference in mean plasma concentration between sexes H 0 : There is on interaction of sex and hormone treatment on the mean plasma concentration Why not just use one-way ANOVA with for levels? 3

4. How to do a 2-way ANOVA with equal replications Calculating means Calculate cell means: Calculate the total mean (grand mean) Calculating treatment means 4

5. How to do a 2-way ANOVA with equal replications Calculating general Sum of Squares Calculate total SS: Calculate the cell SS Calculating treatment error SS 5

6. How to do a 2-way ANOVA with equal replications Calculating factor Sum of Squares Calculating factor A SS: Calculating factor B SS Calculating A  B interaction SS A  B interaction SS = cell SS – factor A SS – factor B SS = 4,9005 A  B DF = cell DF– factor A DF – factor B DF = 1 6

7. How to do a 2-way ANOVA with equal replications Summary of calculations 7

8. How to do a 2-way ANOVA with equal replications Hypothesis test H 0 : There is on effect of hormone treatment on the mean plasma concentration F = hormone MS/within-cell MS = 1386,1125/18,8370 = 73,6 F 0,05(1),1,16 = 4,49 H 0 : There is on difference in mean plasma concentration between sexes F = sex MS/within-cell MS = 3,73 F 0,05(1),1,16 = 4,49 H 0 : There is on interaction of sex and hormone treatment on the mean plasma concentration F = A  B MS/within-cell MS = 0,260 F 0,05(1),1,16 = 4,49 8

9. Visualizing 2-way ANOVA Table 12.2 and Figure 12.1 9

10. 2-way ANOVA in SPSS 10

11. 2-way ANOVA in SPSS 11 Click Add

12. Visualizing 2-way ANOVA without interaction 12

13. Visualizing 2-way ANOVA with interaction 13

14. 2-way ANOVA Random or fixed factor Random factor: Levels are selected at random… Fixed factor: The ’value’ of each levels are of interest and selected on purpose. 14

15. 2-way ANOVA Assumptions Independent levels of the each factor Normal distributed numbers in each cell Equal variance in each cell Bartletts homogenicity test (Section 10.7) s 2 ~ within cell MS; ~ within cell DF The ANOVA test is robust to small violations of the assumptions Data transformation is always an option (see chpter 13) There are no non-parametric alternative to the 2-way ANOVA 15

16. 2-way ANOVA Multiple Comparisons Multiple comparesons tests ~ post hoc tests can be used as in one-way ANOVA Should only be performed if there is a main effect of the factor and no interaction 16

17. 2-way ANOVA Confidence limits for means 95 % confidence limits for calcium concentrations on in birds without hormone treatment 17

18. 2-way ANOVA With proportional but unequal replications Proportional replications: 18

19. 2-way ANOVA With disproportional replications Statistical packges as SPSS has porcedures for estimating missing values and correcting unballanced designs, eg using harmonic means Values should not be estimated by simple cell means Single values can be estimated, but remember to decrease the DF 19

20. 2-way ANOVA With one replication Get more data! 20

21. 2-way ANOVA Randomized block design 21

22. 3-way ANOVA 22

23. 3-way ANOVA H 0 : The mean respiratory rate is the same for all species H 0 : The mean respiratory rate is the same for all temperatures H 0 : The mean respiratory rate is the same for both sexes H 0 : The mean respiratory rate is the same for all species H 0 : There is no interaction between species and temperature across both sexes H 0 : There is no interaction between species and sexes across temperature H 0 : There is no interaction between sexes and temperature across both spices H 0 : There is no interaction between species, temperature, and sexes 23

24. 3-way ANOVA Latin Square 24

25. Exercises 12.1, 12.2, 14.1, 14.2 25